Algorithm to Integrate the Ffowcs Williams-Hawkings Equation on Supersonic Rotating Domain

Abstract

The numerical solution of the Ffowcs Williams -Hawkings equation (Ffowcs Williams, J. E., and Hawkings, D. L., Sound Generation by Turbulence and Surfaces in Arbitrary Motion, Philosophical Transactions of the Royal Society , Vol. A264, No. 1151, 1969, pp. 321 -342) on a rotating supersonic domain is discussed. Based on the emission-surface algorithm, the adopted solver performs the integration on the so-called acoustic domain to avoid the Doppler singularity in the integral kernels. The presence of multiple emission times for the supersonic source points and the particular time evolution of the integration domain force the use of a particular data-e tting procedure on both the geometrical and integral quantities. The algorithm may be used in thenumerical prediction of the quadrupole source term for helicopter rotors operating at a high transonic regime and in the aeroacoustic analysis of the modern propeller blades, rotating at supersonic tip speed. INCE the end of the 1970s, when the importance of nonlin- ear terms was highlighted in the numerical estimation of noise from a high tip speed rotating blade, a change in the development and the application of theoretical and computational methodologies for the aeroacoustic analysis of helicopter rotors has occurred. Be- cause of the requirement for a three-dimensional integration and the presence of the Doppler singularity in the integral kernels, the adoption of the Ffowcs Williams -Hawkings(FW-H) 1 equation for the numerical prediction of the high-speed impulsive (HSI) noise hasalwaysbeenconsidereddife cultandcomputationallyexpensive. Over the past 15 years, interest of the aeroacoustic community has progressively moved toward alternative solution forms, such as the Kirchhoffapproach 2 andthecomputationalaeroacousticsmethods, 3 relegating the acousticanalogy approach to the role of alinear prob- lemsolver.Actually,thecomputationofthenonlineartermsisavery dife cult task. The delocalization of the shock waves off the blade tip arising at Mtip ¸ 0:88 forces the extention of the computing domain beyond the sonic cylinder, where the Doppler factor pre- vents the usual FW -H solvers from achieving a reliable prediction of noise. From a theoretical point of view, the problem may be sim- ply bypassed through the use of the emission-surface algorithms, where the integrals are determined on the blade retarded cone gu- ration, and the Doppler singularity does not appear in the integral kernels.Recently,anewmethodhasbeenproposedandsuccessfully implemented, 4 where the calculations proceed forward with respect to time, thus avoiding the solution of the retarded-time equation and the effects of the Doppler singularity. The emission-surface ap- proach rarely has been used: The occurrence of multiple emission timesinthesupersonicregioncausesunconnectedpatchestoappear, which temporarily link together into a single domain, following a time evolution, which is very dife cult to numerically model. Very interesting results concerning the integration on the acous- tic surface have been published by Wells 5;6 and Wells and Han. 7 In these papers the need for determining, at each time step, a new computational grid with a clustering of the integration points along some particular critical radii was recognized, and very smooth sig- natureswereobtainedinthedeterminationoftheFW -Hlinearterms from a rectangular rotor blade and a propfan-type blade rotating at supersonic tip speed. Wells provides a detailed discussion of the mathematical aspects of the problem and points out the important